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Unformatted text preview: Math 131 October 22, 2003 Midterm Exam Name: exam. Be sure to read each question
(1. Show your work neatly and clearly—
uce your score. Partial credit may be
t to the right answer. You
ses, and straightedges. No There are 100 possible points on this
carefully and answer the question aske
without justiﬁcation may red
given for a correct approach even if you don’t ge
may use models of polyhedra, calculators, compas notes or textbooks are allowed. answers 1. A Greek Adventure You leave your apartment one rn Greece. You are asked for your passpo
forced to demonstrate are promptly arrested. At your trial, you are your knowledge of Greek culture by doing the following two tasks. (Re
member, this is your only chance to save yourself from the wretched
dungeon of geometric illiteracy!)
the point P on the line E, construct a line per— assing through P. Give a full proof of why your orning and ﬁnd yourself in ancient
rt, which you don’t have, and (a) (20 points) Given
pendicular to E and p
construction will work. “If a point is on the perpendicular m the endpoints of
t?!” (b) (15 points) The prosecutor says
bisector of a line segment, then it is equidistant fro the segment.” Then he demands “What do you think about tha If he is telling the truth, give a proof. If it is possible that he is lying, give a counterexample to his claim. 360 °
2. (20 points) Is a regular 255gon constructible? How about a angle? Why or why not? 3. (15 points) Your uncle Bob is a master stonemason. One night at
the local bar he brags that he can tile anything with anything. The
bartender bets him he can’t tessellate the high school gym using only
regular pentagons. Who will win thebet? How do you know? (You
may assume that Bob works infinitely fast and the gym ﬂoor is a plane.)
Extra credit (5 points) for explaining whether or not Bob can t_il_e the gym with regular pentagons. R is the Golden Ratio, compute (exactly) R2 — R. 4. (15 points) If
learly state the correct [If you are not conﬁdent of your answer, you can c
value of R for partial credit] hedron has twice as many edges as
how many edges does it have? For 1e of such a polyhedron. 5. (15 points) Suppose a convex poly
it has vertices. If it has ten faces, extra credit (5 points), give an examp ...
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